Theorem fvprif 13641

Description: The value of the pair function at an element of 2o. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
fvprif	|- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )

Proof of Theorem fvprif

Step	Hyp	Ref	Expression
1		fvpr0o 13639	. . . . 5 |- ( A e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
2	1	3ad2ant1 1049	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
3	2	adantr 276	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
4		simpr 110	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> C = (/) )
5	4	fveq2d 5694	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) )
6	4	iftrued 3644	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> if ( C = (/) , A , B ) = A )
7	3, 5, 6	3eqtr4d 2281	. 2 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
8		fvpr1o 13640	. . . . 5 |- ( B e. W -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
9	8	3ad2ant2 1050	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
10	9	adantr 276	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
11		simpr 110	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> C = 1o )
12	11	fveq2d 5694	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) )
13		1n0 6695	. . . . . 6 |- 1o =/= (/)
14	13	neii 2422	. . . . 5 |- -. 1o = (/)
15	11	eqeq1d 2247	. . . . 5 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( C = (/) <-> 1o = (/) ) )
16	14, 15	mtbiri 686	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> -. C = (/) )
17	16	iffalsed 3647	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> if ( C = (/) , A , B ) = B )
18	10, 12, 17	3eqtr4d 2281	. 2 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
19		elpri 3728	. . . 4 |- ( C e. { (/) , 1o } -> ( C = (/) \/ C = 1o ) )
20		df2o3 6692	. . . 4 |- 2o = { (/) , 1o }
21	19, 20	eleq2s 2333	. . 3 |- ( C e. 2o -> ( C = (/) \/ C = 1o ) )
22	21	3ad2ant3 1051	. 2 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( C = (/) \/ C = 1o ) )
23	7, 18, 22	mpjaodan 810	1 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   (/)c0 3520   ifcif 3635   {cpr 3706   <.cop 3708   ` cfv 5372   1oc1o 6670   2oc2o 6671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-1o 6677  df-2o 6678

This theorem is referenced by: (None)